To solve the quadratic equation 4x^2 - x - 3 = 0, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

In this equation, a = 4, b = -1, and c = -3. Plugging these values into the formula, we get:

x = (1 ± √((-1)^2 - 4(4)(-3))) / 2(4)
x = (1 ± √(1 + 48)) / 8
x = (1 ± √49) / 8
x = (1 ± 7) / 8

This gives us two possible solutions:

x = (1 + 7) / 8 = 8 / 8 = 1
x = (1 - 7) / 8 = -6 / 8 = -3/4

Therefore, the solutions to the equation 4x^2 - x - 3 = 0 are x = 1 and x = -3/4.